Theorem tfr2b 7379

Description: Without assuming ax-rep 4699, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.)

Hypothesis

Ref	Expression
tfr.1	⊢ 𝐹 = recs(𝐺)

Assertion

Ref	Expression
tfr2b	⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))

Proof of Theorem tfr2b

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordeleqon 6880	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		eqid 2610	. . . . 5 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
3	2	tfrlem15 7375	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4		tfr.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
5	4	dmeqi 5247	. . . . 5 ⊢ dom 𝐹 = dom recs(𝐺)
6	5	eleq2i 2680	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ dom recs(𝐺))
7	4	reseq1i 5313	. . . . 5 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴)
8	7	eleq1i 2679	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
9	3, 6, 8	3bitr4g 302	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
10		onprc 6876	. . . . . 6 ⊢ ¬ On ∈ V
11		elex 3185	. . . . . 6 ⊢ (On ∈ dom 𝐹 → On ∈ V)
12	10, 11	mto 187	. . . . 5 ⊢ ¬ On ∈ dom 𝐹
13		eleq1 2676	. . . . 5 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
14	12, 13	mtbiri 316	. . . 4 ⊢ (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
15	2	tfrlem13 7373	. . . . . 6 ⊢ ¬ recs(𝐺) ∈ V
16	4	eleq1i 2679	. . . . . 6 ⊢ (𝐹 ∈ V ↔ recs(𝐺) ∈ V)
17	15, 16	mtbir 312	. . . . 5 ⊢ ¬ 𝐹 ∈ V
18		reseq2 5312	. . . . . . 7 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = (𝐹 ↾ On))
19	4	tfr1a 7377	. . . . . . . . . 10 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
20	19	simpli 473	. . . . . . . . 9 ⊢ Fun 𝐹
21		funrel 5821	. . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹)
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ Rel 𝐹
23	19	simpri 477	. . . . . . . . 9 ⊢ Lim dom 𝐹
24		limord 5701	. . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹)
25		ordsson 6881	. . . . . . . . 9 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On)
26	23, 24, 25	mp2b 10	. . . . . . . 8 ⊢ dom 𝐹 ⊆ On
27		relssres 5357	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
28	22, 26, 27	mp2an 704	. . . . . . 7 ⊢ (𝐹 ↾ On) = 𝐹
29	18, 28	syl6eq 2660	. . . . . 6 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = 𝐹)
30	29	eleq1d 2672	. . . . 5 ⊢ (𝐴 = On → ((𝐹 ↾ 𝐴) ∈ V ↔ 𝐹 ∈ V))
31	17, 30	mtbiri 316	. . . 4 ⊢ (𝐴 = On → ¬ (𝐹 ↾ 𝐴) ∈ V)
32	14, 31	2falsed 365	. . 3 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
33	9, 32	jaoi 393	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
34	1, 33	sylbi 206	1 ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  dom cdm 5038   ↾ cres 5040  Rel wrel 5043  Ord word 5639  Oncon0 5640  Lim wlim 5641  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  recscrecs 7354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355

This theorem is referenced by:  ordtypelem3  8308  ordtypelem9  8314